by Kirill Krasnov and Roberto Percacci

The geometric unification of gravity with the other interactions is not currently a popular subject. It is generally believed that a unified theory can only be constructed once a quantum theory of gravity is available. The purpose of this CQG+ contribution is to advocate that it may be fruitful and even necessary to reverse the logic: instead of “quantising in order to unify” it may be necessary to “unify in order to quantise”. If the latter perspective is correct, our current approaches to quantum gravity would be similar to trying to understand the quantum theory of electricity and magnetism separately before they were unified in Maxwell’s theory.

There are several arguments for such a change of priorities.

Kirill Krasnov, Professor of Mathematical Physics, University of Nottingham

Roberto Percacci, Associate Professor of Theoretical Particle Physics at SISSA, Italy

The first is that “top-down” quantum theories have a hard time making contact to the real world. String theory is viewed by many as an already unified quantum theory of gravity, but its relation to phenomenology requires contrived constructions, and even so remains incomplete. Other approaches such as LQG are largely based on the assumption that gravity can be quantised on its own. They are also far from making contact with the real world. Having a classical unified theory to start with may avoid some of these difficulties.

Another argument comes from the SM itself. It was spectacularly confirmed by the discovery of the Higgs boson in 2012. The data coming from the LHC show no signs of new physics and suggest that the SM may be valid all the way to the Planck scale. Indeed, at the theoretical level, the SM remains a valid effective field theory up to Planck energies. Moreover, the SM contains hints of new physics happening near the Planck scale, and thus “knows” about the latter:
(i) coupling constants of the strong and electroweak interactions approximately meet one or two orders of magnitude below the Planck energy [F. Wilczek, Phys.Today 54N11 (2001) 12];
(ii) the self-coupling of the Higgs boson flows to zero near the Planck scale [M. Shaposhnikov and C. Wetterich, Phys.Lett. B683 (2010) 196-200]. This puts the next energy frontier near the Planck scale, and makes the question of how particle physics fits together with gravity the key question to drive the future progress in this field. Once again, this suggests that the time may be ripe to revisit the problem of geometric unification with gravity.

Finally, we would like to point out that every revolution in physics brings with it a new type of geometry. In this sense one can say that physics is geometry. Indeed, gauge theories of which Maxwell’s unified theory of electricity and magnetism was a forerunner bring in the geometry of fibre bundles. General Relativity was based on Riemannian geometry. Quantum mechanics puts at the forefront the subject of symplectic geometry. It is thus only natural to expect that the sought unification of gravity with other interactions will be based on a new type of geometry. It is possible that that the new geometric structures uncovered by the search of a classical unified theory may also lead to progress on the front of quantum gravity.

The idea that matter fields could play an important role in the quantisation of gravity has always been quite popular among particle physicists. In the 70’s and 80’s it motivated a revival of the Kaluza-Klein approach to unification, enhanced by the rich structures of supergravity. These ideas did not work, and were superseded by developments in string theory. However, there exist alternative ways of unifying gravity and gauge interactions that do not require new spacetime dimensions.

It is interesting that these theories were briefly discussed by Einstein himself in his work on the unified theory. However, Einstein was not aware of crucial pieces of the unification puzzle, and these theories still remain largely unexplored.

Ultimately, the unified theory of all interactions will have to be a quantum theory, but the search for a new classical geometric framework for unification may be an important overlooked ingredient. It is possible that the history of physics has already made a full circle and that the time has come to take the problem of unification seriously once again. Our topical review paper, which was just published by CQG, reviews most attempts at classical unification with gravity, focusing on the possibilities that have been least explored.

The full CQG review article can be read here.

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